"However, at the end of "Mudd's Women", Kirk picks up 6 crystals. That leaves us with these viable options:
1. We see 3 circuits go out on screen and there were two more that went out off screen during the rescue operation. There are 6 Active crystal circuits in addition to bypass circuits that do not need crystals.
2. The Enterprise already lost 2 crystals prior to "Mudd's Women" due to wear and tear or other circumstances. We see them lose the next 3 during the rescue attempt on screen. There are 6 Active crystal circuits in addition to bypass circuits that do not need crystals.
3. Kirk picked up 4 crystals and 2 additional as spares. There are 4 Active crystal circuits in addition to bypass circuits that do not need crystals.
As to "The Alternative Factor", "The Paradise Syndrome" and "Elaan of Troyius", there is nothing explicit on how many crystals the ship needs to run at full power. It might be four as seen in "The Alternative Factor" or six like possibly in "Mudd's Women"."
I had also thought theory # 1 might be an option, but Scotty makes it clear that the 3 the audience saw burn out were the 3 that did
SPOCK: What's wrong?
SCOTT: I don't know, sir. With those three lithium crystals gone
SPOCK: It'll take longer on battery power.
MCCOY: Never trusted this.
Theory # 3 makes sense to me. We've seen how quickly the Enterprise
lost 3 crystals just by extending the shields for an apparently short amount of time and I wonder what happened to their crystals in Where No Man Has Gone Before.
It's almost a lucky coincidence they are close to Rigel 12 (less than two days away) so better get some spares while you are being presented with such a rare opportunity.
There is an interesting piece of dialogue where Kirk possibly and totally misunderstood Spock's question:
KIRK: Stand by, Mister Spock. We're coming aboard with the lithium crystals.
SPOCK [OC]: How many coming, Captain?
CHILDRESS: Eve will stay, today at least. We, er, we want to talk.
EVE: You've got someone up there called the Enterprise.
KIRK: Two of us, Mister Spock. Come on, Mister Mudd.
I imagine Mr. Spock raising both of his eyebrows because to me that doesn't seem to be the answer he was looking for...